Strict Self-Assembly of Discrete Self-Similar Fractal Shapes
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916270526431232 |
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| author | Becker, Florent |
| author_facet | Becker, Florent |
| contents | This paper gives a (polynomial time) algorithm to decide whether a given Discrete Self-Similar Fractal Shape can be assembled in the aTAM model.In the positive case, the construction relies on a Self-Assembling System in the aTAM which strictly assembles a particular self-similar fractal shape, namely a variant $K^\infty$ of the Sierpinski Carpet. We prove that the aTAM we propose is correct through a novel device, \emph{self-describing circuits} which are generally useful for rigorous yet readable proofs of the behaviour of aTAMs.We then discuss which self-similar fractals can or cannot be strictly self-assembled in the aTAM. It turns out that the ability of iterates of the generator to pass information is crucial: either this \emph{bandwidth} is eventually sufficient in both cardinal directions and $K^\infty$ appears within the fractal pattern after some finite number of iterations, or that bandwidth remains ever insufficient in one direction and any aTAM trying to self-assemble the shape will end up either bounded with an ultimately periodic pattern covering arbitrarily large squares. This is established thanks to a new characterization of the productions of systems whose productions have a uniformly bounded treewidth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08548 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strict Self-Assembly of Discrete Self-Similar Fractal Shapes Becker, Florent Discrete Mathematics This paper gives a (polynomial time) algorithm to decide whether a given Discrete Self-Similar Fractal Shape can be assembled in the aTAM model.In the positive case, the construction relies on a Self-Assembling System in the aTAM which strictly assembles a particular self-similar fractal shape, namely a variant $K^\infty$ of the Sierpinski Carpet. We prove that the aTAM we propose is correct through a novel device, \emph{self-describing circuits} which are generally useful for rigorous yet readable proofs of the behaviour of aTAMs.We then discuss which self-similar fractals can or cannot be strictly self-assembled in the aTAM. It turns out that the ability of iterates of the generator to pass information is crucial: either this \emph{bandwidth} is eventually sufficient in both cardinal directions and $K^\infty$ appears within the fractal pattern after some finite number of iterations, or that bandwidth remains ever insufficient in one direction and any aTAM trying to self-assemble the shape will end up either bounded with an ultimately periodic pattern covering arbitrarily large squares. This is established thanks to a new characterization of the productions of systems whose productions have a uniformly bounded treewidth. |
| title | Strict Self-Assembly of Discrete Self-Similar Fractal Shapes |
| topic | Discrete Mathematics |
| url | https://arxiv.org/abs/2405.08548 |